Series, Curves, and Vectors

The last class met yesterday; the final’s next week. This blog appears to have been a total loss; next time (if ever), I’ll make looking at it part of the course requirements so I’ll at least have a few readers.

Section 11.5: ## 6, 12, 18, 24, 30.
Due Tuesday 6/3.

Due Thursday 5/29.
Section 9.7, #46.
Section 9.8 ##12, 22, 30.
Section 11.4 ##30, 48.

The integral has come up three times this quarter in three different contexts. Maybe I should memorize it. Let .

Then , so (letting and ), integration by parts gives


;
it follows that and so .

(The last part of this computation involves much the trickiest of basic-trig-function integrals; of course one evaluates it by multiplying its integrand, , up-and-down by [and a u-substitution with the same u]).

With this result in hand, we can, for example, determine the area between the curve (a hyperbola) given by and the x-axis, as t varies from 0 to (say): substituting the appropriate formulas (and endpoints) into gives .

The same integral is needed in computing arc length for the Archimedean spiral . I forget the third case (but’ll probably succeed in looking it up soon: watch this space).

The main computation of this post is done (more clearly) in Wikipedia; I only realized this after typing up my version. Somebody wrote the whole thing up in (broken) verse as a blog post.

Section 11.3: 46, 48.
Section 9.7: 20, 28.

There’ve been a couple homeworks not posted here. The current assignment isn’t due until 5/29—i.e., after the exam.

For the “vector” portion of tomorrow’s quiz, everybody should be ready to compute (i) the angle formed by two vectors in and (ii) the “cross product” of vectors in . Asking for this may turn out to be wildly ambitious in light of the fact that the computations were introduced only yesterday. We’ll see.

In the meantime, here’s something I wish I’d included in that introduction. The “determinant” calculation for the cross-product is probably the best way to go in the general case … but one should also know what goes on in the simplest cases. So. Here goes. Consider the standard basis vectors . One easily verifies (from the “determinant” calculation of the text and lecture) that and .

Note that “multiplying” (maybe we should say “crossing”) and in the opposite order changes the sign of the product. This phenomenon is called anti-commutatation, and was an epoch-making discovery (1843, W. R. Hamilton). One also has and as well as and . These same algebraic “laws” obtain in Hamilton’s quaternions. (Our vector-space with cross-product is not equivalent to the quaternions, however; for example , whereas in the quaternions, one has .)

The point of all this for us is that we can express arbitrary vectors as “sums of scalar multiples of basis vectors” (for example ); one can then compute cross-products of arbitrary vector pairs by expressing each vector as such a sum and performing a “formal multiplication” using the anti-commutation properties of the basis vectors and the additonal rule that scalars commute with everything. I’ll have much more to say about this in class.

In a carelessly-chosen quiz problem yesterday, I asked for he circumference for the curve given by . The relevant integral is (of course!) ; this simplifies to (for full credit, alas). The appropriate “trick” for evaluating this integral, which we’d never discussed, is to “rationalize the numerator”: . At this point a small subtlety arises: we’d like to replace with in the numerator … but a moment’s thought reveals that this substitution will make sense only when . So we’ll take advantage of the symmetry of our figure to get ; this easily evaluates (via “u-substitution” with ) to .

.

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Another approach would be to work with (which requires only the more familiar “double angle law”). Wolfram has an animated cardioid. There’s some discussion here.

The Unapologetic Mathematician has recently posted a piece on the Ratio and Root Tests (one of an ongoing series on Analysis [="Calculus Done Right"]). Since we happen to be in the middle of the section of our text about this topic, I couldn’t resist posting this here.

Due Thursday May 1.
Section 9.4: 4, 8, 12, 16, 20.
Section 10.4: 74, 80, 82, 84.
Section 10.5: 8, 30, 48.

The class had Exam I on Thursday and I didn’t post a new homework assignment. The one before that (HW 6) was in the form of an “Exam Review” handout. That’s why I haven’t posted any assignments recently.

But I also haven’t posted anything else … three weeks now. Well, believe it or not, we’re pretty much on target in going over the material. Mostly I think I’ve stayed away mostly because my workload unexpectedly more than doubled recently. But I’ve also had exactly no feedback from my work here so far & it’s kind of a motivation drainer. Anyhow, here are some remarks on recent material.

Section 9.3,That the series converges for and diverges for is, as the text asserts, easily verified by The Integral Test (so easily, in fact, that it seems more reasonable to redo the computation right there in the text rather than quote Theorem Such-and-so and fight the reader by making us flip through all those extra pages). In my opinion it shouldn’t go without saying in this context that the famous Zeta Function is obtained by extending the function to the “punctured complex plane” ( with removed). So a convergent p-series could also be called simply “zeta of p“. A certain conjecture about Zeta (”the Riemann Hypothesis”) happens to be among the most famous unsolved problems in Mathematics.

Section 10.4. Probably it would be a good idea to plot a few Polar Graphs “by hand”; free grids can be found for printout at, for example, this “MathBits.com” page. Meanwhile, in yet another notational quibble, I’m sort of surprised to find that the text uses to develop its formulas; I’d've gone with (as it seems to me that, for example, is easier to work with than the corresponding textbook formula [with and in place of my r and r']). Of course this is a pretty trivial matter; indeed, I’ve decided to follow the textbook’s notation in the lecture notes just as an exercise.